Integrand size = 26, antiderivative size = 297 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^2 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^2 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}} \]
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Time = 0.24 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5952, 5556, 3384, 3379, 3382} \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^2 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^2 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {c x-1}} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5952
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {-1+c x}} \\ & = -\frac {\sqrt {1-c x} \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {-1+c x}} \\ & = -\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c^2 \sqrt {-1+c x}}+\frac {\left (3 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c^2 \sqrt {-1+c x}} \\ & = -\frac {\left (\sqrt {1-c x} \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c^2 \sqrt {-1+c x}}+\frac {\left (3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c^2 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c^2 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c^2 \sqrt {-1+c x}}-\frac {\left (3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c^2 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c^2 \sqrt {-1+c x}} \\ & = -\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^2 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^2 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.58 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 c^2 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]
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Time = 0.60 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+\operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}-3 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+2 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}+2 \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-3 \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}\right )}{32 \left (c x +1\right ) c^{2} \left (c x -1\right ) b}\) | \(252\) |
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\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
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\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x\,{\left (1-c^2\,x^2\right )}^{3/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
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